| L(s) = 1 | − 5-s + 7-s + 2·11-s − 3·13-s + 2·17-s − 19-s − 6·23-s + 25-s + 4·29-s − 8·31-s − 35-s − 37-s − 4·41-s + 4·43-s − 2·47-s − 6·49-s + 8·53-s − 2·55-s + 6·59-s − 13·61-s + 3·65-s + 3·67-s − 11·73-s + 2·77-s + 7·79-s + 8·83-s − 2·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.603·11-s − 0.832·13-s + 0.485·17-s − 0.229·19-s − 1.25·23-s + 1/5·25-s + 0.742·29-s − 1.43·31-s − 0.169·35-s − 0.164·37-s − 0.624·41-s + 0.609·43-s − 0.291·47-s − 6/7·49-s + 1.09·53-s − 0.269·55-s + 0.781·59-s − 1.66·61-s + 0.372·65-s + 0.366·67-s − 1.28·73-s + 0.227·77-s + 0.787·79-s + 0.878·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 11 T + p T^{2} \) | 1.97.l |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.968596669935869582926883034978, −7.37127424619381903362920180287, −6.63645752510821096908932557129, −5.77648636467018822135102799842, −4.98444121205818135633036821947, −4.19746743118559710194353822518, −3.49264144556112768849490000135, −2.40803024641310212061115919423, −1.42534531461745386641514684394, 0,
1.42534531461745386641514684394, 2.40803024641310212061115919423, 3.49264144556112768849490000135, 4.19746743118559710194353822518, 4.98444121205818135633036821947, 5.77648636467018822135102799842, 6.63645752510821096908932557129, 7.37127424619381903362920180287, 7.968596669935869582926883034978
